Existence of parabolic minimizers to the total variation flow on metric measure spaces

Abstract

We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{X},d,\hat{I}¼)$ satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $\hat{I}\text{c◯}\times (0,T)$ with $\hat{I}\text{c◯}\subset \mathcal{X}$ an open set and $T>0$, we prove existence in the weak parabolic function space $L_w^1(0,T;\mathrm{BV}(\hat{I}\text{c◯}))$. In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.

Accepted, to appear in Manuscripta Math